1. We discussed a Cournot game with incomplete information in class, where firm 1 has a constant marginal cost c, which is known to firm 2, but firm 1 only knows that firm 2's marginal cost is either a high cost c# with probability p or a low cost of with probability 1 - p The demand function is P= a - ( where P is the market price and ( is the total production, i.e. Q = q, + 92, and q, is the production quantity of Firm /. We have computed the Bayesian Nash equilibrium (BNE) for this game as below: Firm 1's BNE strategy: q = p . (a -2c + c#)/3 + (1- p) . (a -2c + cl)/3. Firm 2's BNE strategy: 42 = qall = (a -2c#+c)/3 + (1-p)(c#-c)/6 and 481 = (a -2cl+ c)/3 - p(c#-()/6 Here the superscript B stands for Bayesian game. In this question, we want to study how the payoffs of the firms are affected by the asymmetric information. a. If firm 2's cost is high, compute the payoffs of firms 1 and 2 in the Bayesian Nash equilibrium. Compare them with the payoffs of the two firms in the Nash equilibrium of the complete information Cournot game (the marginal costs are common knowledge, and c, = c#). Which firm is better off in the incomplete information game, which firm is worse off? b. If firm 2's cost turns out to be low, compute the payoffs of the two firms in the Bayesian Nash equilibrium. Compare them with the Nash equilibrium of the complete information Cournot game where co = c. Which firm is better off in the incomplete information game, which firm is worse off? 2. Two partners simultaneously decide to invest in a project. If partner i invests x (x, 2 0) then the revenue from investment to partner i is either a" x with probability p or a Lx with probability (1 - p), and a # > al. Notice that a # and a L apply for both partners. The cost of investment to partner i is given by x2 + x x, . This cost structure is common knowledge to both partners. However, the revenue information is asymmetric: before the partners invest, partner 1 knows whether a # is realized or a L is realized, but partner 2 does not. What is the Bayesian Nash equilibrium of this game? 3. Consider the second price, sealed bid auction of n players ( > 2). For player /, his/her true valuation of the good is v, which is a private information. Player /'s strategy b, (v,) is the bid that he/she submitted, which is a function of the true valuation. Argue that always bidding lower than the true value, i.e., b (v )